There are three principal methods that may be used for onboard computation of the azimuth and elevation of the Earth relative to the spacecraft. Each of these methods has certain advantages and disadvantages which apply in different cases. Moreover, the accuracy obtained with a particular method depends on the orbit.
Direct expansion of the azimuth and elevation can be determined as functions of time based on curve fits to a ground computer's processing of a "true" azimuth and elevation. However, direct polynomial expansion of the azimuth and elevation is the least flexible of the known methods, but it is simpler to implement onboard a spacecraft, and spacecraft processor throughput requirements are minimal.
The method of general perturbations involves series expansions that are developed for the classical orbital elements, which are then used with Kepler's equation to provide the position of the spacecraft at the desired time. The Kepler orbit generator is also potentially capable of accounting for secular perturbations but, in practice, the number and complexity of the corrections may become unwieldy in some cases. Accordingly, as the need for improved pointing accuracy requires consideration of additional corrections, the limits of onboard processing limit the value of this method.
In a method of special perturbations, the spacecraft dynamical equations of motion are integrated directly to determine the instantaneous position. However, the use of traditional three position/three velocity integration to propagate orbital equations of motion causes singular elements that cause special processing requirements for certain conditions, is not normalized to permit self-testing, checking that proper integration is performed without runaway, and requires complicated transformation from inertial to orbital coordinates. In addition, it requires solutions with trigonometric functions or of non-linear equations that require substantial processing time and additional code. As a result, this method forces use of the simplest form representative of the orbital state and does not permit adjustments for special case phenomena.
As a result, prior known implementations of the onboard computation of azimuth and elevation of the earth relative to the spacecraft have employed some form of either the direct expansion method or the method of general perturbations described above. Hughes spacecraft such as Intelsat VI and HSA10 use a simple form of the direct expansion method to derive the platform pointing profile that maintains earth-center pointing, using sun as the inertial reference. The method generates adjustments to the desired sun-earth angle using a linear fit approximation to the orbit. However, the resulting accuracy of this method is limited. Polynomial fits of higher order could be used to increase accuracy but have the disadvantages that a large number of parameter uploads to the spacecraft are required and a great deal of ground processing is also required. In addition, such calculations are not robust since small errors in the coefficients may result in large pointing errors, and they are not very flexible as different orbits may require different forms for the azimuth and elevation expansions. Furthermore, the calculations are difficult to implement if the attitude precession is not small.
Two Hughes HS-601 spacecraft for EHF and UHF communication employ simple onboard Kepler orbit propagators to generate orbital position and derive spacecraft-to-earth target vector. The EHF spacecraft propagates mean anomaly based on the difference between system time and epoch time, given periodic upload of the standard orbital elements, the mean anomaly rate, and the orbital drift rate. This implementation uses a power series expansion solution for Kepler's equation which assumes small orbital eccentricity. The UHF spacecraft assumes a simple elliptical orbit and uses an Adams-Bashforth integration scheme to propagate orbital position and orbit height. The earth line in earth centered inertial (ECI) coordinates is then determined from the orbit location and a predetermined transformation matrix from an ECI frame to an orbit frame centered in the spacecraft.
The accuracy resulting from the Kepler orbit propagators, while improved over the linear approximation, is still limited, especially near perigee for more highly elliptical orbits. Although increased accuracy may be achieved using this approach, the implementation would be computation intensive, involving trigonometric functions and the solution of non-linear equations.